A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System

Chen, Wenbin1; Jing, Jianyu2; Wang, Cheng3; Wang, Xiaoming4,5,6,7

2022-08-01

10.1007/s10915-022-01872-1

JOURNAL OF SCIENTIFIC COMPUTING   Impact Factor & Quartile

0885-7474

1573-7691

In this paper, we propose and analyze a finite difference numerical scheme for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term, explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection and the phase field coupled term, respectively. Such a semi-implicit method gives an intermediate velocity field. Subsequently, a Helmholtz projection into the divergence-free vector field yields the velocity vector and the pressure variable at the next time step. This approach decouples the Stokes solver, which in turn drastically improves the numerical efficiency. The positivity-preserving property and the unique solvability of the proposed numerical scheme is theoretically justified, i.e., the phase variable is always between -1 and 1, following the singular nature of the logarithmic term as the phase variable approaches the singular limit values. In addition, an iteration construction technique is applied in the positivity-preserving and unique solvability analysis, motivated by the non-symmetric nature of the fluid convection term. The energy stability of the proposed numerical scheme could be derived by a careful estimate. A few numerical results are presented to validate the robustness of the proposed numerical scheme.

Cahn-Hilliard-Navier-Stokes system Flory Huggins energy potential Positivity preserving Monotonicity analysis Energy stability Numerical accuracy

SCI ; EI

English

Others

NSFC[11871159,12071090] ; NSF[DMS-2012269] ; Guangdong Provincial Key Laboratory for Computational Science and Material Design[2019B030301001]

Mathematics

Mathematics, Applied

WOS:000812635800001

SPRINGER/PLENUM PUBLISHERS

20222512248367

Approximation algorithms ; Chemical analysis ; Finite difference method ; Iterative methods ; Navier Stokes equations ; Velocity

Mathematics:921 ; Calculus:921.2 ; Numerical Methods:921.6

MATHEMATICS

Web of Science

1.Fudan Univ, Sch Math Sci, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China
2.Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China
3.Univ Massachusetts, Math Dept, N Dartmouth, MA 02747 USA
4.Southern Univ Sci & Technol, Int Ctr Math, Shenzhen 518055, Peoples R China
5.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Peoples R China
6.Southern Univ Sci & Technol, Guangdong Prov Key Lab Computat Sci & Mat Design, Shenzhen 518055, Peoples R China
7.Southern Univ Sci & Technol, Natl Ctr Appl Math Shenzhen, Shenzhen 518055, Peoples R China

Chen, Wenbin,Jing, Jianyu,Wang, Cheng,et al. A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System[J]. JOURNAL OF SCIENTIFIC COMPUTING,2022,92(2).

Chen, Wenbin,Jing, Jianyu,Wang, Cheng,&Wang, Xiaoming.(2022).A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System.JOURNAL OF SCIENTIFIC COMPUTING,92(2).

Chen, Wenbin,et al."A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System".JOURNAL OF SCIENTIFIC COMPUTING 92.2(2022).